Curves with prescribed rational points
Abstract
Abstract Given a smooth curve $$C/{\mathbb {Q}}$$ C / Q with genus $$\ge 2$$ ≥ 2 , we know by Faltings’ Theorem that $$C({\mathbb {Q}})$$ C ( Q ) is finite. Here we ask the reverse question: given a finite set of rational points $$S\subseteq {\mathbb {P}}^n({\mathbb {Q}})$$ S ⊆ P n ( Q ) , does there exist a smooth projective curve $$C/{\mathbb {Q}}$$ C / Q contained in $${\mathbb {P}}^n$$ P n such that $$C({\mathbb {Q}})=S$$ C ( Q ) = S ? We answer this question in the affirmative by providing an effective algorithm for constructing such a curve.
Fields of Study & Subjects
Advanced Numerical Analysis Techniques
Algebraic Geometry and Number Theory
Mathematical Dynamics and Fractals
Publication details
- Journal / source
- Research in Number Theory
- Publisher
- Springer Science and Business Media LLC
- Volume
- 12
- Issue
- 4
- Pages
- 91–91
- Publication date
- 18 September 2026
- Language
- EN